Math and science::Algebra::Aluffi

Polynomial rings

When polynomials are viewed as a type of notational container, they can be seen to have a ring structure.

Polynomial

A polynomial is a combination of "powers" of an indeterminate $x$ with coefficients in a ring $R$ . It is written in the form:

a_{n} x^{n} + . . . + a_{1} x + a_{0}

\sum_{i > 0} a_{i} n^{i}

We require that there is an $n$ above which all coefficients are zero.

Two polynomials are considered equal if [what?].

The meaning of the operation $\cdot$ in $a_{i} \cdot x^{i}$ is not specified in the definition. Furthermore, the meaning of the $+$ operation between elements is also not specified. The definition of equality cements the notational behaviour.

Notation

The set of polynomials with indeterminate $x$ over ring $R$ is denoted as $R [x]$ .

Polynomials as rings

A polynomial $R [x]$ forms a ring (operations on the reverse side). As polynomials form rings, the construction can be nested such that polynomials in indeterminate $y$ over $R [x]$ for a ring. This is denoted as $R [x] [y]$ and $R [x, y]$ .

Polynomial rings can be generalized and be considered instances of [what?] rings.

27.06.2022